Parameter update method, parameter update apparatus, and non-transitory recording medium storing program for parameter update

ABSTRACT

There is provided a method of updating a setting value of a variable parameter, the method including: obtaining a time-series data of control input and a time-series data of control output observed in control with a controller; calculating a value of the variable parameter which minimizes an output value of an evaluation function based on the obtained time-series data of the control input and the control output; and updating the setting value of the variable parameter to the calculated value of the variable parameter. The evaluation function includes a first function part in which a first norm or a second norm changes depending on the value of the variable parameter, and a second function part of which specific frequency band has an amount, by which the output value of the evaluation function is increased, larger than that of any other frequency band.

CROSS REFERENCE TO RELATED APPLICATION

This application is a divisional application of U.S. application Ser.No. 15/279,981 filed on Sep. 29, 2016, which claims priority fromJapanese Patent Application No. 2015-194530 filed on Sep. 30, 2015, thedisclosures of which are incorporated herein by reference in theirentireties.

BACKGROUND Field of the Invention

The present disclosure relates to a parameter update method and aparameter update apparatus.

Description of the Related Art

As conventional automatic adjustment technologies for controllers, thereare known the Fictitious Reference Iterative Tuning (FRIT) approach andthe Virtual Reference Feedback Tuning (VRFT) approach. Thesetechnologies adjust a parameter of the controller so that the parameterapproximates a target response.

SUMMARY

The conventional automatic adjustment technologies, unfortunately,merely adjust the parameter of the controller to obtain a control outputcorresponding to the target response, and thus a control system maybecome unstable. Namely, the automatic adjustment of the controller bythe known automatic adjustment technology may destabilize the controlsystem, thereby dispersing the response.

In view of the above, an object of the present disclosure is to providea technology which is capable of adjusting a parameter of a controllerwhile maintaining the stability of a control system.

According to an aspect of the present disclosure, a parameter updatemethod relates to a method of updating a setting value of a variableparameter in a controller. The controller is configured to control acontrol target based on a control input, which is calculated accordingto a predetermined transfer function with the variable parameter basedon a deviation between a control output and a target value of thecontrol output.

The parameter update method according to the aspect of the presentdisclosure includes: obtaining time-series data of the control input andtime-series data of the control output observed in control with thecontroller; calculating a value of the variable parameter whichminimizes an output value of an evaluation function based on theobtained time-series data of the control input and the control output;and updating the setting value of the variable parameter in thecontroller to the calculated value of the variable parameter.

The evaluation function includes a first function and a second function.The first function may include the time-series data of the control inputand the control output, the predetermined transfer function with thevariable parameter, and a model function indicating a reference value ofthe control output corresponding to the target value.

As an example, the first function may be configured to output a norm ofa difference between an observation value of the control output at eachtime identified from the time-series data and a reference value of thecontrol output corresponding to the value of the variable parameter.Thus, the first function may be configured so that the norm of thedifference changes depending on the value of the variable parameter. Aninverse model of the predetermined transfer function can obtain a targetvalue corresponding to an observation value of the control input. Thereference value of the control output in the evaluation function may bean output value of a model function obtained by inputting the targetvalue corresponding to the observation value of the control input to theabove-described model function. Accordingly, the reference value of thecontrol output may be configured to change depending on the value of thevariable parameter.

As another example, the first function may be configured to output anorm of a difference between an observation value of the control inputat each time identified from the time-series data and a reference valueof the control input corresponding to the value of the variableparameter. Thus, the first function may be configured so that the normof the difference changes depending on the value of the variableparameter. An inverse model of the model function can obtain a targetvalue corresponding to an observation value of the control output. Thereference value of the control input in the evaluation function may be acontrol input obtained by inputting, to the predetermined transferfunction, a deviation between the observation value of the controloutput and a target value obtained from the inverse model of the modelfunction. Thus, the reference value of the control input may beconfigured to change depending on the value of the variable parameter.

The second function may be a function in which the output value of theevaluation function corresponding to the norm of the differenceincreases as a change in the predetermined transfer function caused by achange in the value of the variable parameter is greater, and of whichspecific frequency band has an amount, by which the output value isincreased, larger than that of any other frequency band.

According to the aspect of the present disclosure, since the evaluationfunction includes the second function, the value of the variableparameter which minimizes the output value of the evaluation function iscalculated so as not to greatly change the characteristics of thetransfer function. In particular, according to the aspect of the presentdisclosure, the value of the variable parameter which minimizes theoutput value of the evaluation function is calculated so as not togreatly change the characteristics of the transfer function in thespecific frequency band.

Thus, the parameter update method of the present disclosure can adjustthe variable parameter of the transfer function set in the controller ina range in which the characteristics of the frequency band, which ismore likely to become unstable, are not changed greatly. In other words,the parameter update method of the present disclosure can adjust thevariable parameter of the controller while maintaining the stability ofthe control system.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic cross-sectional view of surroundings of a sheetconveyance mechanism in an image forming system.

FIG. 2 is a block diagram of an electrical configuration of the imageforming system.

FIG. 3 is a block diagram depicting a position controller and aparameter update unit.

FIG. 4 is an illustrative diagram of the FRIT approach.

FIG. 5 is a flowchart indicating a method of setting an evaluationfunction.

FIG. 6 is a Bode diagram of a transfer function to be mounted on acontroller.

FIG. 7 is a diagram indicating a white-noise signal before and after afilter process.

FIG. 8 is a diagram indicating a frequency spectrum of the white-noisesignal before and after the filter process.

FIG. 9 is a flowchart indicating a tuning process to be executed by amain unit (parameter update unit).

FIG. 10A is a graph indicating an observation value and a targetresponse of a control output, and FIG. 10B is a partial enlarged diagramof FIG. 10A.

FIG. 11 is a graph indicating observation values of the control outputbefore and after a parameter update according to a first embodiment.

FIG. 12 is a Bode diagram indicating frequency characteristics of thetransfer function before and after the parameter update according to thefirst embodiment.

FIGS. 13A to 13C are partial enlarged diagrams of FIG. 12.

FIG. 14 is an illustrative diagram of the VRFT approach.

FIG. 15 is a graph indicating observation values of the control outputbefore and after a parameter update according to a second embodiment.

FIG. 16 is a Bode diagram indicating frequency characteristics of thetransfer function before and after the parameter update according to thesecond embodiment.

FIGS. 17A and 17B are partial enlarged Bode diagrams of FIG. 16.

DESCRIPTION OF THE EMBODIMENTS

Hereinafter, an explanation will be made about embodiments of thepresent disclosure with reference to drawings. The following describesan example in which technologies related to a parameter update methodand a parameter update apparatus of the present disclosure are appliedto an image forming system which jets liquid droplets of ink on a sheetQ to form an image thereon. The technologies related to the presentdisclosure, however, are not limited to the application to the imageforming system, and they can be applied to various systems.

First Embodiment

An image forming system 1 depicted in FIG. 1 is formed as an ink-jetprinter. The image forming system 1 includes a recording head 10 whichjets liquid droplets of ink on the sheet Q to form an image thereon. Therecording head 10 is carriage on a carriage 21. The carriage 21reciprocates in a main scanning direction (the normal direction of asheet surface of FIG. 1) perpendicular to a sheet conveyance directionto move the recording head 10 in the main scanning direction.

Similar to known ink-jet printers, the image forming system 1intermittently conveys the sheet Q below the recording head 10 by apredetermined amount by repeatedly performing the conveyance control ofthe sheet Q in which the sheet Q is conveyed by the predetermined amountand then stopped. When the sheet Q is being stopped, the carriage 21moves in the main scanning direction and the recording head 10 jetsliquid droplets of ink. Accordingly, an image in the main scanningdirection is formed on the sheet Q every time the sheet Q is conveyedintermittently. The image forming system 1 forms an image across thesheet Q by performing such an operation repeatedly.

The sheet Q is conveyed from the upstream to the downstream of theplaten 39 positioned below the recording head 10 by receiving the actionof force of a conveyance roller 31 and a discharge roller 35. The sheetconveyance direction is orthogonal to rotating shafts of the conveyanceroller 31 and the discharge roller 35. The conveyance roller 31 isdisposed upstream of the platen 39 to face a driven roller 32. Thedischarge roller 35 is disposed downstream of the platen 39 to face adriven roller 36.

The conveyance roller 31 is rotated by a PF motor 71 constructed of adirect-current motor (DC motor). Rotating the conveyance roller 31 in astate that the sheet Q is nipped between the conveyance roller 31 andthe driven roller 32 conveys the sheet Q downstream. The dischargeroller 35 is connected to the conveyance roller 31 via a connectionmechanism 38. The discharge roller 35 rotates synchronously with theconveyance roller 31 by receiving the power from the PF motor 71 via theconveyance roller 31 and the connection mechanism 38. Rotating thedischarge roller 35 in a state that the sheet Q is nipped between thedischarge roller 35 and the driven roller 36 conveys the sheet Q, whichis conveyed from the side of the conveyance roller 31 along the platen39, further downstream. The conveyance roller 31, the driven roller 32,the discharge roller 35, the driven roller 36, the connection mechanism38, and the platen 39 constitute a conveyance mechanism 30 (see FIG. 2)of the sheet Q.

As depicted in FIG. 2, the image forming system 1 of this embodimentincludes a main unit 40, a communication interface 50, a feed unit 60, asheet conveyance unit 70, and a recording unit 90. The main unit 40,which includes a CPU 41, a ROM 43, a RAM 45, and NVRAM 47, controls theimage forming system 1 integrally. The CPU 41 executes processingaccording to programs stored in the ROM 43. The RAM 45 is used as aworking memory when the CPU executes processing. The NVRAM 47 is anelectrically data-rewritable nonvolatile memory, and it stores a datawhich is required to be maintained after the image forming system 1 isturned off.

When the main unit 40 receives a data to be printed from an externaldevice 5 via the communication interface 50, the main unit 40 inputscommands to the feed unit 60, the sheet conveyance unit 70, and therecording unit 90 so as to form an image based on the data to be printedon the sheet Q.

The feed unit 60 conveys the sheet Q from an unillustrated feed tray tothe nipping position of the sheet Q by use of the conveyance roller 31and the driven roller 32 in accordance with the command from the mainunit 40. The sheet conveyance unit 70 intermittently conveys the sheet Qsupplied from the feed unit 60 to an image forming position below therecording head 10 in accordance with the command from the main unit 40.

When the sheet conveyance unit 70 stops the intermittent conveyance ofthe sheet Q, the recording unit 90 makes the recording head 10 jetliquid droplets of ink based on the data to be printed while moving thecarriage 21 in the main scanning direction, according to the commandfrom the main unit 40. This forms an image in the main scanningdirection on the sheet Q. The recording unit 90 includes the recordinghead 10 and a carriage movement mechanism 20 which enables the carriage21 carrying the recording head 10 to reciprocate in the main scanningdirection.

Specifically, when the main unit 40 receives the data to be printed, themain unit 40 makes the feed unit 60 and the sheet conveyance unit 70convey the leading edge of the sheet Q to the image forming positionbelow the recording head 10, and then the main unit 40 makes therecording unit 90 move the carriage 21 (recording head 10) in the mainscanning direction and perform the image forming operation for the sheetQ. After that, the main unit 40 sets a target locus r(t) during theintermittent conveyance on the sheet conveyance unit 70, and the mainunit 40 makes the sheet conveyance unit 70 convey the sheet Q accordingto the target locus r(t) by a predetermined amount. Further, the mainunit 40 makes the recording unit 90 perform the image forming operationfor the sheet Q.

After the image forming operation, the main unit 40 makes the sheetconveyance unit 70 re-execute the conveyance operation of the sheet Qaccording to the target locus r(t). At the time of stoppage of theconveyance operation, the main unit 40 makes the recording unit 90 movethe carriage 21 in the main scanning direction and re-execute the imageforming operation for the sheet Q. The main unit 40 makes the sheetconveyance unit 70 and the recording unit 90 execute the processingalternatingly, thereby forming the image based on the data to be printedon the sheet Q.

More specifically, the sheet conveyance unit 70 includes the conveyancemechanism 30, the PF motor 71, a motor drive circuit 73, a rotaryencoder 75, a signal processing circuit 77, and a controller 80. Asdescribed above, the conveyance mechanism 30 is a conveyance mechanismof the sheet Q including the conveyance roller 31, the driven roller 32,the discharge roller 35, the driven roller 36, the connection mechanism38, and the platen 39. The conveyance mechanism 30 conveys the sheet Qby the rotations of the conveyance roller 31 and the discharge roller 35upon receipt of the power from the PF motor 71.

The PF motor 71 is driven by the motor drive circuit 73 to rotate theconveyance roller 31. The motor drive circuit 73 drives the PF motor 71by applying, to the PF motor 71, a drive current (or a drive voltage)corresponding to a control input u to be input from the controller 80.

The rotary encoder 75 is constructed similarly to known rotary encoders.The rotary encoder 75 outputs, as encoder signals, pulse signals oftwo-phases including an A-phase pulse signal and a B-phase pulse signalaccording to the rotation of the conveyance roller 31. The rotaryencoder 75 is provided, for example, in a power transmission pathbetween the PF motor 71 and the conveyance roller 31. The signalprocessing circuit 77 measures a rotation position y and a rotationvelocity v of the conveyance roller 31 on the basis of each encodersignal to be input from the rotary encoder 75, and the signal processingcircuit 77 inputs the measured rotation position y and rotation velocityv of the conveyance roller 31 to the controller 80.

The controller 80 includes a target command unit 81, a positioncontroller 83, and an observation unit 85. The target command unit 81inputs, to the position controller 83, a target value r of the rotationposition y at each time t from a start point of control according to thetarget locus r(t) set by the main unit 40.

As depicted in FIG. 3, the position controller 83 inputs, to the motordrive circuit 73, a control input u corresponding to a deviation e=(r−y)between the target value r and the rotation position y obtained from thesignal processing circuit 77. P indicated in FIG. 3 represents atransfer function P of a control target, and it satisfies a relationalexpression y=P·u. In particular, the position controller 83 converts thedeviation e into the control input u according to a preset transferfunction C(ρ) (u=C(ρ)·e). Through that operation, the positioncontroller 83 feedback-controls the rotation position y with referenceto the target value r.

The target locus r(t) which is set by the main unit 40 during theintermittent conveyance of the sheet Q represents a position locus alongwhich the sheet Q is conveyed from a position at the start of control toa target stop position corresponding to the predetermined amount and thestoppage of the sheet Q is maintained. As an example, the target locusr(t) represents a locus by which a value corresponding to the positionat the start of control (r=0) is monotonically increased to a valuecorresponding to the target stop position (r=y_(s)) and the target valuer is maintained at the value r=y_(s). As another example, the targetlocus r(t) may be a constant value r=y, corresponding to the target stopposition. The position controller 83 drives and controls the PF motor 71according to the target locus r(t), thereby rotating the conveyanceroller 31 to convey the sheet Q by the predetermined amount.

The transfer function C(ρ) of the position controller 83 used in thecalculation of the control input u has a variable parameter ρ set by themain unit 40, and the variable parameter ρ is adjusted by the main unit40. The main unit 40 functions as a parameter update unit 49. Executionof the program by the CPU 41 makes the parameter update unit 49 update asetting value of the variable parameter ρ based on an observational dataobtained from the observation unit 85.

The observation unit 85 collects, for a certain period, based on thecommand input from the main unit 40, a pair of the control input u asthe control input which is observed when the position controller 83controls the PF motor 71 based on the target locus r(t) and the rotationposition y as the control output which is measured by the signalprocessing circuit 77, and the observation unit 85 provides, to the mainunit 40, a time-series data of the control input u and a time-seriesdata of the rotation position y as the observational data.

The main unit 40 updates the setting value of the variable parameter ρbased on the observation data. The main unit 40 functions as theparameter update unit 49 to update the setting value of the variableparameter ρ for the position controller 83 when a specified condition issatisfied, such as when the image forming system 1 is turned on. Theupdated setting value can be stored in the NVRAM 47.

Subsequently, an explanation will be made about a principle of update ofthe variable parameter p. The parameter update unit 49 of thisembodiment calculates an appropriate value of the variable parameter ρfor the position controller 83 by calculating the value of the variableparameter ρ which minimizes the next evaluation function J1, and theparameter update unit 49 sets the calculated appropriate value in theposition controller 83.

J1=∥Y−Y*∥ ² +∥G(ρ)∥²  Formula 5

Y*=T·(C(ρ)⁻¹ ·U+Y)  Formula 6

As represented in the following formulas, Y included in the evaluationfunction J1 is a vectorial representation of the observation value(rotation position y) of the control output represented by theobservational data, and U is a vectorial representation of theobservation value (control input u) of the control input represented bythe observational data. In the following, the observed rotation positiony is also referred to as an observation value y of the control outputand the observed control input u is also referred to as an observationvalue u of the control input.

Y=(y[1],y[2], . . . ,y[M])

U=(u[1],u[2], . . . ,u[M])

A component y[k] (1≤k≤M) of Y and a component u[k] (1≤k≤M) of Ucorrespond to the control output and the control input observed at eachtime, respectively. Each of y[k] and u[k] is sequenced in thecorresponding one of the vectors Y, U in the order of time. In otherwords, y[k] and u[k] are “time series data”.

T included in the evaluation function J1 is a model functionrepresenting a reference value Y_(d) of the control output (rotationposition y) corresponding to the target value r. Namely, the modelfunction T is a transfer function in which the reference value Y_(d) isrepresented by a relational expression Y_(d)=T·r. A first member of theevaluation function J1 corresponds to a norm of the difference between avectorial representation Y of the observation value y and a vectorialrepresentation Y* of the reference value Y_(d). Here, the L2 norm isadopted as the norm.

The first member of the evaluation function J1 is an evaluation functionfollowing the known FRIT approach. Adjusting the variable parameter ρchanges the observation value y. Here, it is troublesome to provide theobservation value y for each value of the variable parameter ρ forsearching the variable parameter ρ in which the norm of the differencebetween the vectorial representation Y of the observation value y andthe vectorial representation Y* of the reference value Y_(d) is minimum.In the FRIT approach, by converting the reference value Y_(d)=T·r of thecontrol output to a reference value T·r*(ρ) of the control outputaccording to the variable parameter ρ, the variable parameter ρ in whichthe norm of the difference between the vectorial representation Y of theobservation value y and the vectorial representation Y* of the referencevalue Y_(d) is minimum can be searched by using the observation value yindependent of the variable parameter ρ. r*(ρ) is a pseudo-target valuecorresponding to the target value r which satisfies a relationalexpression r*(ρ)=C⁻¹(ρ)·u+y.

Inputting the pseudo-target value r*(ρ) into the position controller 83obtains the observation value y of the control output represented by theobservational data, as understood from FIG. 4. The reference valueT·r*(ρ) of the control output according to the variable parameter ρ isobtained by inputting the pseudo-target value r*(ρ) into the modelfunction T.

Specifically, the first member of the evaluation function J1 correspondsto a norm of the difference between a vectorial representation Y of theobservation value y and a vectorial representation Y* of the referencevalue T·r*(ρ) of the control output according to the variable parameterρ. In other words, the first member of the evaluation function J1corresponds to a norm of the difference between the observation value yof the control output at each time and a reference value T·(C⁻¹(ρ)·u+y)of the control output corresponding to the value of the variableparameter ρ. The value of the variable parameter ρ which minimizes thefirst member of the evaluation function J1 corresponds to a value of thevariable parameter ρ by which the reference value Y_(d)=T·r is observedas the control output corresponding to the target value r.

In the conventional FRIT approach, the variable parameter ρ is searchedwithout consideration of the stability of a control system. Thus,according to the knowledge of the inventor of the present disclosure,applying the FRIT approach to the control system with a high-ordercomplex transfer function C(ρ) makes the control system unstable.

A second member of the evaluation function J1 adopted in this embodimentis provided to prevent the control system from becoming unstable. Afunction G(ρ) provided for the second member of the evaluation functionJ1 increases the norm ∥G(ρ)∥ as the change in the transfer function C(ρ)caused by the change in the value of the variable parameter ρ isgreater, and a specific frequency band of the function G(ρ) has anamount, by which the norm ∥G(ρ)∥ is increased, larger than those ofother frequency bands. The specific frequency band is a frequency bandin which the adjustment of the variable parameter ρ is more likely todestabilize the control system. The band in which the control system ismore likely to be destabilized is specified by examining the open-loopcharacteristics of the control system, as is well known in the art.

In this embodiment, the function G(ρ) is defined by a function F(ρ)·W(G(ρ)=F(ρ)·W). Here, a weighing vector W can be defined by representing,in a vectorial representation similar to Y and U, a time-series data ofa signal with frequency components most of which are in the specificfrequency band. The signal with frequency components most of which arein the specific frequency band is generated, for example, by inputting awhite noise signal to a filter having the specific frequency band as apassband and other frequency bands as stopbands. As is well known in theart, the white noise signal is a signal with components which areuniformly distributed in all frequency bands, that is, a signal with aflat frequency spectrum. The weighing vector W is represented, forexample, by the following formula by use of a transfer function H of thefilter and a value w [k] of the white noise signal at each time (1≤k≤M).

W=H·(w[1],w[2], . . . ,w[M])

The function F(ρ) includes, as elements, an initial transfer functionCn=(ρ_(ini)) which is obtained by assigning an initial value ρ_(ini) ofthe variable parameter ρ to the transfer function C(ρ) and the transferfunction C(ρ) with the variable parameter ρ. In the function F(ρ), thenorm increases as the ratio or difference between the transfer functionC(ρ) and the initial transfer function Cn is greater. The initial valueρ_(ini) of the variable parameter ρ may be a factory value of thevariable parameter ρ which is not updated by the parameter update unit49. In particular, the function F(ρ) may be any one of the followingfunctions: F1(ρ), F2(ρ), F3(ρ), and F4(ρ) or a combination thereof.

$\begin{matrix}{{F\; 1(\rho)} = \left( {1 - \frac{Cn}{C(\rho)}} \right)} & {{Formula}\mspace{14mu} 7} \\{{F\; 2(\rho)} = \left( {1 - \frac{C(\rho)}{Cn}} \right)} & {{Formula}\mspace{14mu} 8} \\{{F\; 3(\rho)} = {{C(\rho)} - {Cn}}} & {{Formula}\mspace{14mu} 9} \\{{F\; 4(\rho)} = {{Cn} - {C(\rho)}}} & {{Formula}\mspace{14mu} 10}\end{matrix}$

When explained in a general way, in the function F(ρ), the norm of thefunction F(ρ) may indicate a minimum value (e.g., zero) when C(ρ)=Cn issatisfied in a variation range of the variable parameter ρ, and the normof the function F(ρ) may indicate a value greater than the minimum valuewhen C(ρ)=Cn is not satisfied in the variation range of the variableparameter ρ. It is preferred that the function F(ρ) have no minimumpoint except for a point indicating the minimum value in the variationrange of the variable parameter ρ.

The following describes, for ease of explanation, an example in whichthe function F1(ρ) is used as the function F(ρ) of this embodiment and∥F1(ρ)·W∥² is set as the second member of the evaluation function J1.The evaluation function J1 of this example is represented by thefollowing formula.

$\begin{matrix}{{J\; 1} = {{{Y - {T \cdot \left( {{{C(\rho)}^{- 1} \cdot U} + Y} \right)}}}^{2} + {{\left( {1 - \frac{Cn}{C(\rho)}} \right) \cdot W}}^{2}}} & {{Formula}\mspace{14mu} 11}\end{matrix}$

As described above, the evaluation function J1 includes, as elements,the observation values Y, U of the control input and the control output,the model function T, the transfer function C(ρ) of the controller(position controller 83), and the weighing vector W. Of the elements, adesigner is able to determine the transfer function C(ρ) of thecontroller and the model function T according to a targeted controlresponse (T·r).

It is preferred that the designer set the variable parameter ρ of thetransfer function C(ρ) by arranging the variable parameter ρ in thenumerator of C(ρ)⁻¹ and by not arranging the variable parameter ρ in thedenominator of C(ρ)⁻¹. The reason thereof is that a least-squareapproach which is widely used to solve a minimization problem can not beadopted when the variable parameter ρ is arranged in the denominator ofthe evaluation function J1. In this embodiment, the variable parameter ρis indicated by a single symbol. The variable parameter ρ, however, maybe a set of variable parameters ρ₁, ρ₂, . . . ρ_(N).

For example, assuming that the transfer function C is defined by thefollowing formula by use of coefficients A₀, A₁, A₂, A₃, A₄, A₅, A₆ andcoefficients B₁, B₂, B₃, B₄, B₅, B₆. In the following formula, s is aLaplace operator.

$\begin{matrix}{C = \frac{{A_{6}s^{6}} + {A_{5}s^{5}} + {A_{4}s^{4}} + {A_{3}s^{3}} + {A_{2}s^{2}} + {A_{1}s^{1}} + A_{0}}{{B_{6}s^{6}} + {B_{5}s^{5}} + {B_{4}s^{4}} + {B_{3}s^{3}} + {B_{2}s^{2}} + {B_{1}s^{1}}}} & {{Formula}\mspace{14mu} 12}\end{matrix}$

In that case, the transfer function C(ρ) can be defined by setting thecoefficients B₁, B₂, B₃, B₄, B₅, B₆ to the variable parameter ρ={ρ₁, ρ₂,ρ₃, ρ₄, ρ₅, ρ₆}.

$\begin{matrix}{{C(\rho)} = \frac{{A_{6}s^{6}} + {A_{5}s^{5}} + {A_{4}s^{4}} + {A_{3}s^{3}} + {A_{2}s^{2}} + {A_{1}s^{1}} + A_{0}}{{\rho_{6}s^{6}} + {\rho_{5}s^{5}} + {\rho_{4}s^{4}} + {\rho_{3}s^{3}} + {\rho_{2}s^{2}} + {\rho_{1}s^{1}}}} & {{Formula}\mspace{14mu} 13}\end{matrix}$

After determining the transfer function C(ρ), the initial valuep=ρ_(ini), and the model function T, the designer designs the weighingvector W and sets it in the evaluation function J1 in accordance with adesign procedure in FIG. 5. The evaluation function J1 to be used in theparameter update unit 49 is determined, accordingly.

At first, the designer can obtain a frequency characteristic diagram ofthe transfer function C(ρ) which is an update (tuning) target of thevariable parameter ρ (S110). In particular, the designer can obtain aBode diagram. Then, the frequency band, which is more likely to becomeunstable by the adjustment of the variable parameter ρ and in which thechange in frequency characteristics by the adjustment should be reduced,is determined by comparing a gain and a phase in the frequencycharacteristic diagram (S120). In the following, the above frequencyband is referred to as a reduction band.

According to the known determination method, the transfer function C(ρ)is stable provided that, in a Bode diagram showing the open-loopcharacteristics, the value of the gain obtained when the phase passes−180 degrees is lower than 0 dB and the value of the phase obtained whenthe gain passes 0 dB is smaller than −180 degrees.

Assuming that the initial transfer function Cn=C(ρ_(ini)) has thefrequency characteristics following the Bode diagram depicted in FIG. 6.The Bode diagram depicted in FIG. 6 relates to a transfer function Cnwhich is obtained by assigning the initial value ρ=ρ_(ini) to thetransfer function C(ρ) defined by the formula 13. In that case, thephase value is close to −180 degrees in a state that the frequency isclose to a value ω₁. Thus, it can be judged that the change in frequencycharacteristics by the adjustment of the variable parameter ρ should bereduced in a band with the value ω₁ or more. Namely, it can be judgedthat the reduction band is the band with the value ω₁ or more.

The designer can determine a filter which has the reduction band as thepassband, as a filter to be used for generation of the weighing vectorW, according to the judgement result in S120 (S130). When it is judgedthat the reduction band is the frequency band with the value ω₁ or more,for example, a cutoff frequency can be set as the value ω₁ and ahighpass filter which has the band with the value ω₁ or more as thepassband can be determined as the filter to be used for generating theweighing vector W. The transfer function H of the highpass filter can berepresented, for example, by the following formula. A value ω_(c)corresponds to the cutoff frequency.

$\begin{matrix}{H = \left( \frac{s}{s + \omega_{c}} \right)^{10}} & {{Formula}\mspace{14mu} 14}\end{matrix}$

Then, the designer generates the white noise signal through softwareprocessing by a device or computer (S140) and inputs the white noisesignal to a filter in which the passband is in the reduction banddetermined in S130 and the stopband is on the outside of the reductionband, thereby producing a time-series data of the white noise signalafter passing through the filter (S150). The designer can determine thesignal intensity of the white noise signal experimentally so that thesecond member of the evaluation function J1 has a greater influence onthe evaluation function J1 than the first member.

In FIG. 7, the white noise signal after passing the highpass filter isdepicted by a solid line and the white noise signal before passing thehighpass filter is depicted by a broken line. In FIG. 8, a frequencyspectrum of the white noise signal after passing the highpass filter isdepicted by a solid line and the white noise signal before passing thehighpass filter is depicted by a broken line. As understood from FIG. 8,the white noise signal after passing the highpass filter indicates aweak signal intensity in the frequency band with less than the value ω₁,and it has frequency components most of which are in the frequency band(the reduction band) with not less than the value ω₁.

The designer can define the evaluation function J1 by setting avectorial representation H·(w[1], w[2], . . . , w[M]) of the time-seriesdata produced in S150 as the weighing vector W (S160). The weighingvector W can be defined as a M-dimensional vector having the same numberof elements as Y, U. The defined evaluation function J1 can beincorporated in a program which makes the main unit 40 work as theparameter update unit 49, as a function with variables Y, U, p.

The program which makes the main unit 40 work as the parameter updateunit 49 can be designed as a program which makes the main unit 40execute a tuning process depicted in FIG. 9. As an example, the mainunit 40 can execute the tuning process depicted in FIG. 9 regularly orevery time the image forming system 1 is turned on. Executing the tuningprocess regularly includes, for example, execution of the tuning processevery time the number of sheets printed exceeds a predetermined amountand execution of the tuning process every time a given time elapses. Asanother example, the main unit 40 can execute the tuning processdepicted in FIG. 9 every time a user inputs an execution command. Theuser can input the execution command through the external device 5 or auser interface (not depicted) provided in the image forming system 1.

When starting the tuning process, the main unit 40 inputs a command tothe sheet conveyance unit 70 so that the sheet conveyance unit 70performs test conveyance of the sheet Q (S210). The main unit 40 setsthe target locus r(t) in the controller 80 and makes the positioncontroller 83 execute the control of the PF motor 71 based on the targetlocus r(t) and the transfer function C(ρ). The main unit 40 stores thesetting value of the variable parameter ρ in the NVRAM 47 and sets thesetting value in the position controller 83. Before performing the testconveyance, the main unit 40 can input a command to the feed unit 60 sothat the feed unit 60 supplies the sheet Q to the conveyance mechanism30.

The observation unit 85 operates according to the command input from themain unit 40, collects a pair of a control input u, which is observedwhen the position controller 83 controls the PF motor 71 based on thetarget locus r(t) during the test conveyance, and a rotation position ymeasured by the signal processing circuit 77, and provides, to the mainunit 40, a time-series data of the pair of the control input u and therotation position y, as the observational data.

The main unit 40 obtains the observation data provided from theobservation unit 85 (S220). Then, the main unit 40 determines whether ornot a control error is large based on the observation data (S230). Themain unit 40 determines, based on the observation data, whether or notthe observation value y in a steady-state in which the target value r(t)reaches the value r=y, corresponding to the target stop position andr=y, is maintained, is within a predetermined acceptable range (see FIG.10B) on the basis of the target stop position. The main unit 40determines whether the control error is large, accordingly.

The position controller 83 continues to calculate the control input ubased on e=r−y after the target value r(t) reaches r=y₅. Thus, at firstit seems that the operation of the position controller 83 adjusts theconveyance roller 31 and the position of the sheet Q to the target stopposition. But in fact, when the conveyance roller 31 and the sheet Q arestopped once, the load acting on the conveyance roller 31, such asstatic frictional force, increases. The sheet Q will not move unlessdrive force overcoming the increased load acts on the conveyance roller31. This keeps the sheet Q stopped, even when the sheet Q is not in thetarget stop position.

When the observation value y is not within the acceptable range, themain unit 40 determines that the control error is large. Other thanthat, the main unit 40 determines that the control error is small. InFIG. 10A, a locus of the observation value y obtained when the controlerror is large is depicted by a broken line, and the target response(T·r) is depicted by a solid line. FIG. 10B is a partial enlargeddiagram of FIG. 10A (an area surrounded by a two-dot chain line in FIG.10A). The model function T corresponding to the target responseindicated in FIG. 10B is as follows.

$\begin{matrix}{T = \left( \frac{10^{4}}{s + 10^{4}} \right)^{4}} & {{Formula}\mspace{14mu} 15}\end{matrix}$

According to FIG. 10B, the observation value y in the steady-state isnot within the acceptable range of which center is the target response(T·t). In that case, the main unit 40 determines that the control erroris large (S230: Yes).

When the main unit 40 determines that the control error is large, theprocess proceeds to S240. When the main unit 40 determines that thecontrol error is small (S230: No), the main unit 40 judges that thevariable parameter ρ requires no update, completing the tuning process.

In S240, the main unit 40 assigns the observation value u of controlinput and the observation value y of control output indicated by theobservation data obtained in S220 to the evaluation function J1, andcalculates the value of the variable parameter ρ which minimizes theevaluation function J1 after the assignment. In particular, the value ofthe variable parameter ρ which minimizes the evaluation function J1 iscalculated based on the least-square approach.

After that, the main unit 40 sets the calculated value of the variableparameter ρ in the position controller 83 (S250), thereby updating thesetting value of the variable parameter ρ in the position controller 83.Further, the calculated value of the variable parameter ρ is stored, asthe setting value of the variable parameter ρ, in the NVRAM 47, and thenthe tuning process is completed.

FIG. 11, FIG. 12, and FIGS. 13A to 13C show parameter update resultsusing the transfer function C(ρ) represented by the formula 13 and themodel function T represented by the formula 15. FIG. 11 depicts, similarto FIG. 10B, a locus of the observation value y in the steady-state. InFIG. 11, the observation value y before the variable parameter ρ isupdated by the parameter update unit 49 is depicted by a broken line,and the observation value y after the variable parameter ρ is updated bythe parameter update unit 49 is depicted by a chain line. In FIG. 11,the target stop position is depicted by a solid line. As understood fromFIG. 11, the control accuracy by the position controller 83 is improvedby the update of the variable parameter ρ with the parameter update unit49.

As understood from FIG. 12, the transfer function C(ρ) is updated sothat the effect of the second member of the evaluation function J1 doesnot destabilize the control system. In FIG. 12, the frequencycharacteristics of the transfer function C(ρ) before the variableparameter ρ is updated are depicted by a broken line, the frequencycharacteristics after the variable parameter ρ is updated according tothe evaluation function J1 are depicted by a chain line, and thefrequency characteristics after the variable parameter ρ is updatedaccording to a conventional evaluation function are depicted by a solidline. The conventional evaluation function is a function in which thesecond member is removed from the evaluation function J1.

As understood from FIG. 12 and FIGS. 13A to 13C showing enlarged partsof FIG. 12, the frequency characteristics of the transfer function C(ρ)updated by the conventional evaluation function show that the phase isbelow −180 degrees in a state that the gain is 0 dB. This means that thecontrol system is unstable. FIG. 13A is an enlarged view of thefrequency domain D1 of FIG. 12, FIG. 13B is an enlarged view of thefrequency domain D2 of FIG. 12, and FIG. 13C is an enlarged view of thefrequency domain D3 of FIG. 12. Meanwhile, in this embodiment, thevariable parameter ρ is updated appropriately without destabilizing thecontrol system. Namely, in this embodiment, the variable parameter ρ isupdated while achieving both the stability and the control performance.

In this embodiment, even when the characteristics of the control target(transfer function P) change over time, the variable parameter ρ of theposition controller 83 is appropriately adjusted according to the changewith the passage of time, so that the position controller 83 is updatedto achieve the target response. Thus, the image forming system 1maintains the quality of an image to be formed on the sheet Q at a highlevel.

Second Embodiment

An explanation will be made about the image forming system 1 accordingto the second embodiment. The image forming system 1 of the secondembodiment is formed similarly to the image forming system 1 of thefirst embodiment, except that the evaluation function used in theparameter update unit 49 of the first embodiment is changed from theevaluation function J1 to an evaluation function J2 on the basis of theVRFT approach. Thus, the following selectively describes theconfiguration related to the evaluation function J2.

The parameter update unit 49 of this embodiment calculates anappropriate value of the variable parameter ρ for the positioncontroller 83 by calculating the value of the variable parameter ρ whichminimizes the next evaluation function J2, and sets the calculated valuein the position controller 83.

J2=∥U−U*∥ ² +∥G(ρ)∥²  Formula 16

U*=C(ρ)·(T ⁻¹ ·Y−Y)  Formula 17

Y, U included in the evaluation function J2 are similar to Y, U includedin the evaluation function J1. T included in the evaluation function J2is a model function representing a reference value Y_(d) of the controloutput (rotation position y) corresponding to a target value r, similarto the evaluation function J1. The reference value Y_(d) is representedby the relational expression Y_(d)=T·r.

A first member of the evaluation function J2 is an evaluation functionfollowing the known VRFT approach. According to the VRFT approach, asdepicted in FIG. 14, a value T⁻¹·y, which is obtained by inputting anobservation value y in an inverse model T⁻¹ of the model function T, isused as a pseudo-target value r*. A control input u*=C(ρ)·(T¹·y−y),which is obtained by inputting the target value r* in the controller(position controller 83), is used as a reference value of control input,and the variable parameter ρ is updated to reduce the difference betweenthe reference value of control input and an observation value u ofcontrol input.

The first member of the evaluation function J2 corresponds to a norm ofthe difference between a vectorial representation U of the observationvalue u of control input and a vectorial representation U* of areference value u*=C(ρ)·(T⁻¹·y−y) of control input according to thevariable parameter ρ. In other words, the first member of the evaluationfunction J2 corresponds to a norm of the difference between theobservation value u of control input at each time and the referencevalue u*=C(ρ)·(T¹·y−y) of control input corresponding to the value ofthe variable parameter ρ. Thus, similar to the evaluation function J1,the value of the variable parameter ρ which minimizes the first memberof the evaluation function J2 corresponds to the value of the variableparameter ρ by which the reference value Y_(d)=T·r of control outputindicated by the model function is observed as the control outputcorresponding to the target value r.

In the second embodiment, the evaluation function J2 is defined byadding the second member ∥G(ρ)∥ which is similar to the first embodimentto the evaluation function following the VRFT approach. As described inthe first embodiment, the function G(ρ) provided for the second memberof the evaluation function J2 increases a norm ∥G(ρ)∥ as the change inthe transfer function C(ρ) caused by the change in the value of thevariable parameter ρ is greater. In the function G(ρ), its specificfrequency band has an amount, by which the norm ∥G(ρ)∥ is increased,larger than those of other frequency bands. Similar to the firstembodiment, the specific frequency band corresponds to the reductionband, and the adjustment of the variable parameter ρ is more likely todestabilize the control system in the specific frequency band. Inparticular, the evaluation function J2 is defined by the followingformula.

J2=∥U−U*∥ ² +∥F(ρ)·W∥ ²  Formula 18

The function F(ρ) can be defined similarly to the first embodiment. Theweighing vector W can be set according to FIG. 5. The parameter updateunit 49 can calculate the variable parameter ρ which minimizes theevaluation function J2 by using the evaluation function J2 in S240, andthen set the calculated value in the position controller 83.

When the variable parameter ρ which minimizes the evaluation function J2is calculated by using the least-square approach, the variable parameterρ is required to be arranged in the numerator of the evaluation functionJ2, similar to the evaluation function J1. Although the evaluationfunction J1 includes the transfer function C(ρ) as the inverse modelC⁻¹(ρ), the same is not true of the evaluation function J2. Thus, in thesecond embodiment, the transfer function C(ρ) can be defined by settingthe variable parameter ρ in the numerator of the transfer function C.For example, in the transfer function C indicated in the formula 12, thetransfer function C(ρ) can be defined by setting coefficients A₀, A₁,A₂, A₃, A₄, A₅, A₆ in the variable parameter ρ={ρ₀, ρ₁, ρ₂, ρ₃, ρ₄, ρ₅,ρ₆}, as represented by the following formula.

$\begin{matrix}{{C(\rho)} = \frac{{\rho_{6}s^{6}} + {\rho_{5}s^{5}} + {\rho_{4}s^{4}} + {\rho_{3}s^{3}} + {\rho_{2}s^{2}} + {\rho_{1}s^{1}} + \rho_{0}}{{B_{6}s^{6}} + {B_{5}s^{5}} + {B_{4}s^{4}} + {B_{3}s^{3}} + {B_{2}s^{2}} + {B_{1}s^{1}}}} & {{Formula}\mspace{14mu} 19}\end{matrix}$

FIG. 15, FIG. 16, FIG. 17A, and FIG. 17B show update results of thevariable parameter ρ which are obtained by using the transfer functionC(ρ) and the next evaluation function J2.

$\begin{matrix}{{J\; 2} = {{{U - U^{*}}}^{2} + {{\left( {1 - \frac{C(\rho)}{Cn}} \right) \cdot W}}^{2}}} & {{Formula}\mspace{14mu} 20}\end{matrix}$

Similar to FIG. 10B, FIG. 15 depicts a locus of an observation value yin the steady-state. In FIG. 15, the observation value y before thevariable parameter ρ is updated by the parameter update unit 49 isdepicted by a broken line, the observation value y after the variableparameter ρ is updated according to the evaluation function J2 with theparameter update unit 49 is depicted by a chain line, and the targetstop position is depicted by a solid line. As understood from FIG. 15,the control accuracy by the position controller 83 is improved by theupdate of the variable parameter ρ according to the evaluation functionJ2 with the parameter update unit 49.

As understood from FIG. 16 and FIGS. 17A and 17B depicting enlargedparts of FIG. 16, the transfer function C(ρ) is updated so that theeffect of the second member of the evaluation function J2 does notdestabilize the control system. FIG. 17A is an enlarged view of thefrequency domain D4 of FIG. 16, and FIG. 17B is an enlarged view of thefrequency domain D5 of FIG. 16. In FIG. 16, the frequencycharacteristics of the position controller 83 before the variableparameter ρ is updated is depicted by a broken line, the frequencycharacteristics after the variable parameter ρ is updated according tothe evaluation function J2 is depicted by a chain line, and thefrequency characteristics after the variable parameter ρ is updatedaccording to a conventional evaluation function is depicted by a solidline. The conventional evaluation function is a function in which thesecond member is removed from the evaluation function J2.

When the variable parameter ρ is updated by the conventional evaluationfunction, the control system is destabilized greatly. In the secondembodiment, the variable parameter ρ is updated appropriately withoutdestabilizing the control system. Thus, in the second embodiment, evenwhen the characteristics of the control target (transfer function P)change over time, the variable parameter ρ of the position controller 83is adjusted appropriately according to the change with the passage oftime, so that the position controller 83 is updated to achieve thetarget response similar to the first embodiment.

In the above description, the configuration of the image forming system1 and the configuration of the parameter update unit 49 according toeach of the first and second embodiments are explained. The presentdisclosure, however, is not limited thereto, and may adopt variousmodified embodiments.

For example, the application of the parameter update unit 49 of each ofthe above embodiments is not limited to the control system which conveysthe sheet Q intermittently. For example, the update approach of thevariable parameter ρ based on each of the evaluation functions J1 and J2can be applied to the control system of the feed unit 60 of the imageforming system 1. In addition to this, the update approach of thevariable parameter ρ based on each of the evaluation functions J1 and J2can be applied to any other control system than the image forming system1.

In the above embodiments, the functions of the parameter update unit 49are achieved by allowing the main unit 40 to execute the programs. Thepresent disclosure, however, is not limited thereto, and the functionsof the parameter update unit 49 may be achieved by a dedicated hardware.Or, the functions of the parameter update unit 49 may be installed in aninformation processing unit provided independently of the controlsystem. In that case, the information processing unit obtains anobservation data of the control system and calculates an appropriatevalue of the variable parameter ρ.

The function provided in one component in each of the above embodimentsmay be distributed in components. The function provided in componentsmay be integrated in one component. A part of the configurationaccording to each of the above embodiments may be omitted. At least apart of the configuration according to one of the first and secondembodiments may be added to or replaced by the configuration accordingto the other of the first and second embodiments. The present disclosureincludes various embodiments or aspects which are included in thetechnical ideas specified by the following claims.

What is claimed is:
 1. A method of updating a setting value of a variable parameter ρ in a controller configured to control a control target based on a control input u, which is calculated according to a predetermined transfer function C(ρ) with the variable parameter ρ based on a deviation (y−r) between a control output y and a target value r of the control output y (u=C(ρ)·(y−r)), the method comprising: obtaining time-series data of the control input u and time-series data of the control output y observed in control with the controller; calculating a value of the variable parameter ρ which minimizes an output value of an evaluation function based on the obtained time-series data of the control input u and the control output y; and updating the setting value of the variable parameter ρ to the calculated value of the variable parameter ρ, wherein the evaluation function is a function J1 or a function J2 including a vectorial representation U of the time-series data of the control input u and a vectorial representation Y of the time-series data of the control output y, J1=∥Y−Y*∥ ² +∥F(ρ)·W∥ ² Y*=T·(C(ρ)⁻¹ ·U+Y) J2=∥U−U*∥ ² +∥F(ρ)·W∥ ² U*=C(ρ)·(T ⁻¹ ·Y−Y) T included in the evaluation function is a model function indicating a reference value of the control output y corresponding to the target value r, F(ρ) includes an initial transfer function Cn obtained by assigning an initial value of the variable parameter ρ to the predetermined transfer function C(ρ) and the predetermined transfer function C(ρ) with the variable parameter ρ, and in the function F(ρ), the norm increases as a ratio or difference between the predetermined transfer function C(ρ) and the initial transfer function Cn is greater, and the method further comprises: determining the evaluation function by setting time-series data of a signal with frequency components most of which are in a frequency band, in which a change in frequency characteristics of the transfer function C(ρ) from the initial transfer function Cn caused by a change in the value of the variable parameter ρ is reduced, to a weighing vector w included in the evaluation function.
 2. The method according to claim 1, wherein in a case that the evaluation function is determined, a white noise signal is input to a filter of which passband is included in the frequency band in which the change in the frequency characteristics is reduced, to generate, as the signal with the frequency components most of which are in the frequency band in which the change in the frequency characteristics is reduced, time-series data of the white noise signal after passage through the filter, and a vectorial representation of the time-series data is set to the weighing vector W.
 3. A parameter update apparatus configured to update a setting value of a variable parameter ρ in a controller configured to control a control target based on a control input u, which is calculated according to a predetermined transfer function C(ρ) with the variable parameter ρ based on a deviation (y−r) between a control output y and a target value r of the control output y (u=C(ρ)·(y−r)), the apparatus comprising: an obtaining unit configured to obtain time-series data of the control input u and time-series data of the control output y observed in control with the controller; a calculation unit configured to calculate a value of the variable parameter ρ which minimizes an output value of an evaluation function based on the time-series data of the control input u and the control output y obtained by the obtaining unit; and an update unit configured to update the setting value of the variable parameter ρ to the calculated value by the calculation unit, wherein the evaluation function is a function J1 or a function J2 including a vectorial representation U of the time-series data of the control input u and a vectorial representation Y of the time-series data of the control output y, J1=∥Y−Y*∥ ² +∥F(ρ)·W∥ ² Y*=T·(C(ρ)⁻¹ ·U+Y) J2=∥U−U*∥ ² +∥F(ρ)·W∥ ² U*=C(ρ)·(T ⁻¹ ·Y−Y) T included in the evaluation function is a model function indicating a reference value of the control output y corresponding to the target value r, F(ρ) includes an initial transfer function Cn obtained by assigning an initial value of the variable parameter ρ to the predetermined transfer function C(ρ) and the predetermined transfer function C(ρ) with the variable parameter ρ, and in the function F(ρ), the norm increases as a ratio or difference between the predetermined transfer function C(ρ) and the initial transfer function Cn is greater, and a weighing vector w included in the evaluation function is a vectorial representation of time-series data of a signal with frequency components most of which are in a frequency band in which a change in frequency characteristics of the predetermined transfer function C(ρ) from the initial transfer function Cn caused by a change in the value of the variable parameter ρ is reduced. 